WASHINGTON (CNN) -- Congressional Democrats vowed Wednesday to pressure Republican lawmakers to join them in an effort to overridePresident Bush's veto of a bill that would expand a popular children's health insurance program.

House Speaker Nancy Pelosi says she will work to find votes needed to override the president's veto.

The bill, which would have spent $35 billion over five years, would have doubled the number of children eligible for the State Children's Health Program, its supporters said.

House Speaker Nancy Pelosi said she would try to get the 15 additional Republican votes she said she needed to overturn Bush's veto, noting that "2-to-1 Republican voters support SCHIP and oppose the president's veto."

"It's very sad that the president has chosen to veto a bill that would provide health care to 10 million American children for the next five years. It is a value that is shared by the American people across the board," Pelosi said.

Bush exercised the veto at 10 a.m. ET before leaving the White House for a trip to Lancaster, Pennsylvania, to discuss the federal budget and taxes.

Speaking in Pennsylvania, Bush said he vetoed the bill because it was a step toward "federalizing" medicine and inappropriately expanded the program beyond its focus on helping poor children.Watch Congress regroup after the veto »

"I believe in private medicine, not the federal government running the health care system. I do want Republicans and Democrats to come together to support a bill that focuses on the poorer children," the president said, adding the government's policy should be to help people find private insurance.

The House of Representatives voted 222 - 197 to postpone the vote to override the president's veto until October 18. The voting was mostly along party lines vote, with one Democrat voting against the motion.

Congress sent the legislation on the State Children's Health Insurance Program, or SCHIP, to the White House on Tuesday after the Senate voted 67-29 last week to expand the program.

It appears Congress lacks the votes to overturn Bush's veto. Though 67 votes in the 100-person Senate would suffice to override a veto, the 265-159 House vote on September 25 is short of the two-thirds majority needed. View Bush's and previous presidents' veto records »

Rep. Adam Putnan of Florida, chairman of the House Republican conference, said he was confident the president's veto will be sustained.

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"The fact that they are delaying this for two weeks reflects that they're more interested in running radio and TV ads for two weeks than they are in expanding children's health insurance coverage as quickly as possible," Putnam added.

Putnam argued that "the longer the details of this bill hang out there" the better it is for those who oppose it and "time is on our side."

The Democratic Congressional Campaign Committee announced that on Monday it would launch a campaign involving radio ads, automated phone calls and emails targeting eight Republicans who voted against the SCHIP expansion.

After the veto was announced, Democrats quickly took to the floors of the Senate and House of Representatives to condemn the veto of the bill that received bipartisan support.

"I think that this is probably the most inexplicable veto in the history of the country. It is incomprehensible. It is intolerable. It's unacceptable," said Sen. Edward Kennedy, D-Massachusetts, who pleaded with Republicans to help overturn the veto.

House Democrats also were quick to compare the bill's $7 billion annual cost to the money spent each month on the Iraq war.

"The president and Republicans in Congress say that we can't afford this bill, but where were the fiscal conservatives when the president demanded hundreds of billions of dollars for the war in Iraq?" asked Rep. Jan Schakowsky, D-Illinois.

Some House Republicans, however, said Bush was right to veto the bill.

"The public can see that we're playing more political 'gotcha' than we are at really solving problems," said Rep. Todd Akin, R-Missouri, who said the legislation contained "all of these little hidden gizmos, among other things that we're going to provide health care to the children of illegal immigrants."

Akin also said the bill would have led to "a massive expansion of, basically, 'Hillary' socialized medicine," a reference to Sen. Hillary Clinton of New York and her unsuccessful health care efforts as first lady in the 1990s.

Democrats denied the bill would provide coverage to illegal immigrants.

Sen. Orrin Hatch of Utah was among 18 Republicans who split from the president when the bill came up in the Senate.

"It's very difficult for me to be against a man I care so much for," he told his colleagues on the Senate floor before the vote. "It's unfortunate that the president has chosen to be on what, to me, is clearly the wrong side of this issue."

A Washington Post-ABC News poll conducted September 27-30 found 72 percent of those surveyed support an increase in spending on the program, with 25 percent opposed. The poll's margin of error was 3 percentage points.

Bush and many Republicans contend the program's original intent would be changed under the bill.

The program gives coverage to parents who make too much to qualify for Medicaid but not enough to buy private insurance for their children. Critics have said their concern is that parents might be prompted to drop private coverage for their children to get cheaper coverage under the bill.

The veto is the fourth of Bush's presidency. After not using his veto power at all during his first four years, the president has vetoed three other bills in his second term: two on stem-cell research legislation and one on a war funding bill with a Democratic timetable for withdrawing U.S. troops from Iraq. E-mail to a friend